invitation to computer science 6 th edition chapter 2 the algorithmic foundations of computer...

Invitation to Computer Science 6th Edition

Chapter 2

The Algorithmic Foundations of Computer Science

Invitation to Computer Science, 6th Edition

Objectives

In this chapter, you will learn about:

• Representing algorithms

• Examples of algorithmic problem solving

2


Introduction

• Chapter 1 – Introduced algorithms and algorithmic problem

solving

• This chapter– Develops more fully the notions of algorithm and

algorithmic problem solving

3


Representing Algorithms

• Pseudocode– Natural language: used to express algorithms– Problems using natural language to represent

algorithms• Natural language can be extremely verbose

• Lack of structure makes it difficult to locate specific sections of the algorithm

• Natural language is too “rich” in interpretation and meaning

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Figure 2.1 The Addition Algorithm of Figure 1.2 Expressed in Natural Language

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Representing Algorithms(continued)

• High-level programming language– Examples: C++, Java– Problem with using a high-level programming

language for algorithms• During the initial phases of design, we are forced to

deal with detailed language issues

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Figure 2.2 The Beginning of the Addition Algorithm of Figure 1.2 Expressed in aHigh-Level Programming Language

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Pseudocode (continued)

• Natural languages– Not sufficiently precise to represent algorithms

• High-level programming language– During the initial phases of design, we are forced to

deal with detailed language issues

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Pseudocode

• Pseudocode– Used to design and represent algorithms– A compromise between the two extremes of natural

and formal languages

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Figure 1.2 Algorithm for Adding Two m-digit Numbers

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Pseudocode(continued)

• English language constructs modeled to look like statements available in most programming languages

• Steps presented in a structured manner (numbered, indented, and so on)

• No fixed syntax for most operations is required

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Pseudocode(continued)

• Less ambiguous and more readable than natural language

• Emphasis is on process, not notation

• Well-understood forms allow logical reasoning about algorithm behavior

• Can be easily translated into a programming language

12


Sequential Operations

• Basic sequential operations – Computation, input, and output

• Instruction for performing a computation and saving the result– Set the value of “variable” to “arithmetic expression”

• Variable – Storage location that can hold a data value

13


Sequential Operations (continued)

• Pseudocode – Not a precise set of notational rules to be memorized

and rigidly followed

• Input operations – Submit to the computing agent data values from the

outside world that it may then use in later instructions

• Output operations – Send results from the computing agent to the

outside world

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Figure 2.3 Algorithm for Computing Average Miles per Gallon

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Invitation to Computer Science, 6th Edition 16


Conditional and Iterative Operations

• Sequential algorithm– Sometimes called a straight-line algorithm– Executes its instructions in a straight line from top to

bottom and then stops

• Control operations– Conditional and iterative– Allow us to alter the normal sequential flow of control

in an algorithm

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Conditional and Iterative Operations (continued)

• Conditional statements– Ask questions and choose alternative actions based

on the answers– If then else– Example

• if x is greater than 25 then

print x

else

print x times 100

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Figure 2.4 The If/Then/Else Pseudocode Statement

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Figure 2.5 Second Version of the Average Miles per Gallon Algorithm

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• Loop– The repetition of a block of instructions– While statement

• Continuation condition– Determines if statement is true or false

• Infinite loop– Continuation condition never becomes false

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• Examples

– while j > 0 do

set s to s + aj

set j to j – 1

– do

print ak

set k to k + 1

while k < n22


Figure 2.6 Execution of the While Loop23



• Loop example

Step Operation

1 Set the value of count to 1

2 While (count ≤ 100) do step 3 to step 5

3 Set square to (count x count)

4 Print the values of count and square

5 Add 1 to count

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• Pretest loop – Continuation condition is tested at the beginning of

each pass through the loop

• Posttest loop– Continuation condition is tested at the end of the

loop body, not the beginning

• Primitives– Instructions that computing agent understands and is

capable of executing without further explanation

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Figure 2.7 Third Version of the Average Miles per Gallon Algorithm

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Figure 2.8 Execution of the Do/While Posttest Loop

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Figure 2.9 Summary of Pseudocode Language Instructions

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Examples of Algorithmic Problem Solving

• Go Forth and Multiply: Multiply two numbers using repeated addition

• Sequential search: Find a particular value in an unordered collection

• Find maximum: Find the largest value in a collection of data

• Pattern matching: Determine if and where a particular pattern occurs in a piece of text



Examples of Algorithmic Problem Solving

• Example 1: Go Forth and MultiplyGiven 2 nonnegative integer values, a ≥ 0, b ≥ 0, compute and output the product (a 3 b) using the technique of repeated addition. That is, determine the value of the sum a + a + a + . . . + a (b times)

31


Examples of Algorithmic Problem Solving((continued))

• Algorithm outline– Create a loop that executes exactly b times, with

each execution of the loop adding the value of a to a running total

32

• Get values for a and b

• Set the values of count to 0

• Set the value of product to 0

• While (count < b) do– Set the value of product to (product + a)– Set the value of count to (count + 1)

• End of loop

• Print the value of product



Figure 2.10 Algorithm for Multiplication of Nonnegative Values via Repeated Addition

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Modified version of Multiply

• Devise an algorithm that uses a loop to take in a positive integer n and a real number x as the input and computes and print xn . Assume that the computing agent knows how to multiply but do not know how to calculate exponential. Hint: x0 = 1; x1 = x; x2 = x*x; x3= x*x*x; etc. Make sure you stop the algorithm at some point.



Example 2: Looking, Looking, Looking

• Algorithm discovery– Finding a solution to a given problem

• Sequential search– Standard algorithm for searching an unordered list of

values

• Binary Search (extended example)– In computer science, a binary search or half-interval

search algorithm finds the position of a specified value (the input “key”) within a sorted array.

38

Example 2: Looking, Looking, Looking (Continued)

• Task– Find a particular person’s name from an unordered

list of telephone subscribers

• Algorithm outline– Start with the first entry and check its name, then

repeat the process for all entries



Figure 2.11 First Attempt at Designing a Sequential Search Algorithm

40


Example 2: Looking, Looking, Looking (continued))

• For each entry, write a separate section of the algorithm that checks for a match

• Problems of naïve sequential search algorithm– Only works for collections of exactly one size– Duplicates the same operations over and over

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Example 2: Looking, Looking, Looking (continued)

• Correct sequential search algorithm– Uses iteration to simplify the task– Refers to a value in the list using an index (or

pointer)– Handles special cases (such as a name not found in

the collection)– Uses the variable Found to exit the iteration as soon

as a match is found



Figure 2.12 Second Attempt at Designing a Sequential Search Algorithm

43


Figure 2.13 The Sequential Search Algorithm

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Example 2: Looking, Looking, Looking (continued)

• The selection of an algorithm to solve a problem is greatly influenced by the way the data for that problem is organized



Example 3: Big, Bigger, Biggest

• Library– Collection of useful algorithms

• ProblemGiven a value n ≥ 1 and a list containing exactly n unique numbers called A1, A2, . . . , An, find and print out both the largest value in the list and the position in the list where that largest value occurred

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Figure 2.14 Algorithm to Find the Largest Value in a List

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Example 4: Meeting Your Match

• Pattern matching– Process of searching for a special pattern of symbols

within a larger collection of information– Is assisting microbiologists and geneticists studying

and mapping the human genome

51


Example 4: Meeting Your Match (continued)

• Pattern-matching problemYou will be given some text composed of n characters that will be referred to as T1 T2 . . . Tn. You will also be given a pattern of m characters, m ≤ n, that will be represented as P1 P2 . . . Pm. The algorithm must locate every occurrence of the pattern within the text. The output of the algorithm is the location in the text where each match occurred. For this problem, the location of a match is defined to be the index position in the text where the match begins

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• STEP 1– The matching process: T1 T2 T3 T4 T5 … P1 P2 P3

• STEP 2– The slide forward: T1 T2 T3 T4 T5 …– 1-character slide -> P1 P2 P3

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Figure 2.15 First Draft of the Pattern-Matching Algorithm

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Figure 2.16 Final Draft of the Pattern-Matching Algorithm

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• Examples of higher-level constructs– Sort the entire list into ascending order– Attempt to match the entire pattern against the text– Find a root of the equation

• Abstraction– Use of high-level instructions during the design

process

• Top-down design– Viewing an operation at a high level of abstraction

57


Summary

• Algorithm design – A first step in developing an algorithm

• Algorithm design must: – Ensure the algorithm is correct– Ensure the algorithm is sufficiently efficient

• Pseudocode – Used to design and represent algorithms

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Summary (continued)

• Pseudocode – Readable, unambiguous, and able to be analyzed

• Algorithm design – Uses multiple drafts and top-down design to develop

the best solution

• Abstraction – A key tool for good design

59

Test 1

• 4 types questions (11)– Concept: computer science, algorithm, Von

Neumann architecture– Design: using Pseudocode code– Correction: verify the given algorithms– Discussion: be creative & comprehensive

• Close notes & Close books

• Panther Card

• Feb 4th: 3:00-4:15pm

• Assignment 1 due on Feb 4th before 4:15pmInvitation to Computer Science, 6th Edition 60

invitation to computer science 6 th edition chapter 2 the algorithmic foundations of computer...

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